ANALISIS NUMERIK DALAM GEOTEKNIK

By:

Yulinda Sari, ST. M.Eng.

SLOPE/W

SLOPE/W is a software product that uses limit equilibrium theory to compute the factor of safety of earth and rock slopes. APPLICATIONS

SLOPE/W is a powerful slope stability analysis program.

Using limit equilibrium, it has the ability to model heterogeneous soil types, complex stratigraphic and slip surface geometry, and variable pore-water pressure conditions using a large selection of soil models.

Analyses can be performed using deterministic or probabilistic input parameters.

In addition, stresses computed using finite element stress analysis may be used in the limit equilibrium computations for the most complete slope stability analysis available.

The combination of all these features means that SLOPE/W can be used to analyze almost any slope stability problem you will encounter.

APPLICATIONS

1.1. Heterogeneous Slope Overlying Bedrock

1.2. Block Failure Analysis

1.3. External Loads and Reinforcements

1.4. Complex Pore-Water Pressure Condition

1.5. Stability Analysis Using Finite Element Stress

1.6. Dynamic Stability Analysis Using QUAKE Stress

1.7. Probabilistic Stability Analysis

1.1. Heterogeneous Slope Overlying Bedrock

Figure1.1 shows a typical slope stability problem. This specific case has a problematic weak layer located above impenetrable bedrock with a stronger silty clay layer above.The toe of the slope is beneath water, groundwater flows towards the toe, and a tension crack zone has developed at the crest of the slope. The slip surface for this slope is a composite circular arc with straight portions along the bedrock and in the tension crack zone.The Ordinary, Bishop, Janbu Simplified, Spencer, and Morgenstern-Price factors of safety can all be computed for this composite slip surface.

Figure1.1 Heterogeneous Slope Overlying Bedrock

1.2. Block Failure Analysis

Figure1.2 shows a slope stability analysis problem in a system of weak and strong stratigraphy. As shown in the figure, the analysis considers a block failure mode. This analysis also has the toe of the slope beneath water, groundwater flow towards the toe, and a tension crack zone at the crest. A large number of block slip surfaces can be analyzed by specifying a grid of points at the two lower corners of the block. The slip surface is projected upwards from these grid points at a user-specified range of angles.

Figure1.2 Block Failure Mode

1.3. External Loads and Reinforcements

SLOPE/W can calculate the factor of safety for slopes that are externally loaded and reinforced with anchors or geo-fabrics. Figure1.3 shows the SLOPE/W analysis of a slope reinforced using anchors and subjected to external line loads at the crest and a stabilizing berm at the toe.Figure1.3 Example of a External Loads and Reinforcements

1.4. Complex Pore-Water Pressure Condition

Pore-water pressure conditions can be specified in a variety of ways. It may be as simple as a piezometric line or as complex as importing pore-water pressure conditions from a finite element analysis. Another procedure allows you to define the pore-water pressure conditions at a series of points as shown in Figure1.4. The pore-water pressure at the base of each slice is determined from the data points by spline interpolation, (Kriging), techniques.Figure1.4 Example of a Slope with Complex Pore-Water Pressure Condition

1.5. Stability Analysis Using Finite Element Stress

The primary unknown in a slope stability analysis is the normal stress at the base of each slice. An iterative procedure is required to find the normal stress such that the factor of safety is the same for each slice and each slice is in force equilibrium. This iterative procedure can be avoided by importing the slope stresses into SLOPE/W from a SIGMA/W finite element stress analysis. SIGMA/W is a GEO-SLOPE product for finite element stress analyses. The advantage of using finite element computed stresses is that it allows the calculation of the factor of safety for each slice, as well as the overall factor of safety for the slope. Figure1.5 shows a stability analysis performed using SIGMA/W computed stresses.

Figure1.5 Example of a Stability Analysis Using Finite Element Stress

1.6. Dynamic Stability Analysis Using QUAKE Stress

A major feature in Version 5 of SLOPE/W is the ability to integrate with QUAKE/W stress.QUAKE/W is another GEO-SLOPE product for dynamic finite element stress analysis.An earth structure subjected to dynamic or earthquake loadings may be analyzed by QUAKE/W, and the computed stress state integrated into SLOPE/W. The advantage of such an approach is that the factor of safety of the structure during the loading period can be calculated and the permanent deformation of the earth structure as a results of the dynamic loading can be estimated.The following figures show a stability analysis performed using QUAKE/W computed stresses, the computed factor of safety and the estimated deformation as a function of time are also presented.

1.7. Probabilistic Stability Analysis

Some degree of uncertainty is always associated with the input parameters of a slope stability analysis. To accommodate parameter uncertainty in the analysis, SLOPE/W has the ability to perform a Monte Carlo probabilistic analysis. In these cases, each input parameter is specified together with a standard deviation value to define a probability distribution for each input parameter. The standard deviation given for a particular parameter quantifies the degree of uncertainty associated with the parameter. Doing a probabilistic analysis makes it possible to compute a factor of safety probability distribution, a reliability index, and the probability of failure. The probability of failure is defined as the probability that the factor of safety is less than 1.0. The factor of safety is shown as a probability density function in Figure1.6 and as a probability distribution function in Figure1.7.

Figure1.6 Results of Probabilistic Analysis Displayed as a Probability Density Function

Figure1.7 Results of Probabilistic Analysis Displayed as a Probability Distribution Function

Slope Stability Analysis

Analysis Methods

The comprehensive formulation of SLOPE/W allows stability analysis using the following methods:,

- Ordinary (or Fellenius) method,

- Bishop Simplified method,

- Janbu Simplified method,

- Spencer method,

- Morgenstern-Price method,

- Corps of Engineers method, Lowe-Karafiath method,

- generalized limit equilibrium (GLE) method, finite element stress method.

Furthermore, a variety of interslice side force functions can be used with the more mathematically rigorous Morgenstern-Price and GLE methods.

The finite element stress method uses the stress computed from SIGMA/W, (a finite element software product available from GEO SLOPE), to determine a stability factor. All the other methods use the limit equilibrium theory to determine the factor of safety.

The large selection of available analysis methods in SLOPE/W is provided so that you can decide which method suits the problem.

Probabilistic Analysis

SLOPE/W can perform probabilistic slope stability analyses to account for variability and uncertainty associated with the analysis input parameters.

A probabilistic analysis allows you to statistically quantify the probability of failure of a slope using the Monte Carlo method.

The results from all Monte Carlo trials can then be used to compute the probability of failure and generate the factor of safety probability density and distribution functions.

Variability can be considered for material parameters such as unit weight, cohesion and friction angles, pore-water pressure conditions, applied line

Geometry and Stratigraphy

SLOPE/W can be used to model a wide variety of slope geometry and stratigraphy such as multiple soil types, partial submergence in water, variable thickness and discontinuous soil strata, impenetrable soil layers, and dry or water-filled tension cracks.

Tension cracks can be modelled with a specified tension crack line or a maximum slip surface inclination angle.

Slip Surfaces

SLOPE/W uses a grid of rotation centers and a range of radii to model circular and composite slip surfaces.

SLOPE/W also provides block specified slip surface, and fully specified slip surface methods for modelling non-circular slip surfaces.

The following figures illustrate the types of slip surfaces that can be modelled using SLOPE/W.

SLOPE/W provides many options to specify pore-water pressure conditions. Pore-water pressures can be defined as follows:

Pore-water pressure coefficients.

The classic pore water pressure coefficient, ru, which relates the overburden stress to pore-water pressure, can be specified for each soil type.

Piezometric surfaces.

The easiest way to specify pore-water pressure conditions is to define a piezometric surface through the problem domain. For less common, non-hydrostatic situations, such as an artesian sand layer overlain by an clay aquitard, it is possible to define a separate piezometric surface for each soil layer.

Pore-water pressure parameters at specific locations.

If pore-water pressure parameters such as pressure, head, or ru coefficients are known at specific locations within the soil, they can be specified in the model. This feature is useful for incorporating known field data into the analysis or for specifying complex pore-water pressure conditions. Spline interpolation of the specified data is used to calculate the pore-water pressure throughout the problem domain.

Pore-Water Pressures

Finite element computed pore-water pressures.

SLOPE/W has the ability to import pore-water pressure data computed by SEEP/W, VADOSE/W or SIGMA/W, three of GEO-SLOPEs finite element programs. This capability is especially useful for performing slope stability analyses where the groundwater flow conditions are transient and/or significantly affected by the stress state within the soil.

Pore-water pressure contours.

If contours of pore-water pressure distribution are known, perhaps from field observations or some other type of seepage modelling, they can be used to specify the pore-water pressure conditions for a slope stability analysis.

SLOPE/W provides many options to specify pore-water pressure conditions. Pore-water pressures can be defined as follows:

Pore-Water Pressures

Soil Properties

Total and/or effective stress parameters.

The Mohr-Coulomb parameters for cohesion and friction angle are the most common way to model soil shear strength. These parameters can be specified for either total or effective stress conditions in SLOPE/W.

Undrained shear strength.

Undrained soils exhibit no frictional shear resistance. The undrained soil model in SLOPE/W accommodates this by setting the friction angle,f, to zero.

Zero shear strength materials.

For materials which contribute only their weigh but add no shear strength to the system, SLOPE/W provides a zero shear strength material. Examples of zero shear strength materials include ponded water at the toe of a slope and surcharge fills. These materials have zero cohesion, (c=0), and zero friction angle, (f=0).

SLOPE/W provides the following material models to define the soil shear strength.

Soil Properties

Impenetrable materials.

For the purposes of slope stability analyses, material through which a slip surface cannot penetrate are referred to as impenetrable materials. Where a slip surface encounters an impenetrable material such as bedrock, the slip surface continues along the upper boundary of the impenetrable material.

Bilinear failure envelope.

A bilinear Mohr-Coulomb failure envelope is useful for modelling materials that exhibit a change in frictional angle at a particular normal stress.

Increasing cohesive shear strength with depth.

In normally consolidated or slightly overconsolidated soils, cohesion increases with depth. SLOPE/W can accommodate these situations in two ways. The first way is by allowing the cohesive shear strength to vary with the depth below the top layer of the soil. This is useful for the analysis of natural slopes. The second way is by allowing the cohesive shear strength to vary as a function of elevation, independent of the depth from the top layer. This is useful for the analysis of excavated slopes.

SLOPE/W provides the following material models to define the soil shear strength.

Soil Properties

Anisotropic shear strength.

Bedding planes in soil strata result in anisotropic shear strength values for cohesion and friction angle. SLOPE/W has a variety of ways to model anisotropic shear strength parameters, reflecting the variety of engineering practices used throughout the world.

Custom shear strength envelope.

In cases where a linear or bilinear Mohr-Coulomb failure envelope is insufficient for modelling soil shear strength, SLOPE/W provides the capability to specify a general curved relationship between shear strength and normal stress. This is the most general way to specify shear strength.

Shear strength based on normal stress but with an undrained strength maximum.

With this model, the shear strength is based on cohesion and friction angle up to a maximum undrained shear strength. Both cohesion and friction can vary with either depth below ground surface or with elevation above a datum.

Shear strength based on the overburden effective stress.

Soil shear strength in this model is directly related to the overburden effective stress by a specified factor, therefore increasing linearly with depth below the ground surface.

SLOPE/W provides the following material models to define the soil shear strength.

Applied Loads

Several kinds of external applied loads can be modelled using SLOPE/W.

These include surcharge fill and structural loads, toe berm loads, line loads, anchor loads, soil nail loads, geo-fabric loads, and seismic loads.

Formulation

SLOPE/W is formulated in terms of two factor of safety equations. These equations are used to compute the factor of safety based on slice moment and force equilibrium. Depending on the interslice force function adopted, the factor of safety for all the methods can be determined from these two equations.One key difference between the various methods is the assumption regarding interslice normal and shear forces. The relationship between these interslice forces is represented by the parameter l. For example, a l value of 0 means there is no shear force between the slices. A l value that is nonzero means there is shear between the slices.

Comparison with Hand Calculations

The first verification example problem involves the comparison of SLOPE/W solutions to the hand-calculated solutions of a simple slope. The factor of safety from the Ordinary method and the Bishops Simplified method are compared. In both cases, SLOPE/W gives the same results as the hand calculated solutions.

Comparison with Stability Charts

The factor of safety computed by SLOPE/W is 1.34 for both the Bishop Simplified method and the Morgenstern-Price method. This is in close agreement with 1.35 as estimated from the stability charts.

Comparison with Closed Form Solutions

The third verification example problem is to compare the factors of safety computed by SLOPE/W with closed form solutions. Three cases of an infinite slope are considered. In all cases, the factors of safety computed by SLOPE/W are identical to the closed form solutions.

Student Edition

Using the Student Edition :

The Student Edition of each GEO-SLOPE Office product includes a free Student License.

The Student Edition is designed as an aid to learning geotechnical analysis.

It is an ideal teaching tool for university professors both at the undergraduate and graduate levels, and includes documentation and laboratory problems that can be used as a guide for developing class curriculum.

The free Student License is included when you download the software from GEO-SLOPE's web site.

When you run the software, you can select either the Viewer License or the Student License (if you do not already have a full-featured license).

You may freely distribute the Student Edition, provided you adhere to the included license agreement.

Please note that the Student Edition is licensed exclusively for educational and learning purposes and may not be used for professional engineering practice under any circumstances.

For professional engineering use, a full-featured license can be obtained from GEO-SLOPE.

Student Edition

The Student License is a limited version of the software; however, sufficient features are available for learning the basics of geotechnical analysis. The limitations of the SLOPE/W Student License are as follows:

Maximum of 2 soil layers and 1 bedrock layer.

Only the Bishop, Janbu, Ordinary, Spencer, Morgenstern-Price, and

GLE methods can be used.

Only the constant and half-sine interslice force functions can be used.

Only the Grid and Radius slip surface option can be used.

A maximum of 1 piezometric line can be specified.

The only PWP options available are piezometric lines and finite-

element computed pore-water pressure from SEEP/W, SIGMA/W

and QUAKE/W.

Only the Mohr-Coulomb and infinite strength (bedrock) soil models

can be used

Student Edition

No external loading can be applied (e.g., line loads, seismic loads,

or reinforcement)

No pressure lines can be applied

A probabilistic analysis cannot be performed

A tension crack cannot be specified

No strength functions can be used

No air pressure can be specified

The SLOPE/W Student Edition can use finite-element computed

results from SIGMA/W, SEEP/W, and QUAKE/W, as long as the

imported mesh conforms to the Student Edition requirements

(e.g., less than 500 elements)

The Student Edition documentation and laboratory problems

can be downloaded directly from GEOSLOPE's web site at

http://www.geo-slope.com/student.